A101094 a(n) = n*(n+1)*(n+2)*(n+3)*(1+3*n+n^2)/120.

1, 11, 57, 203, 574, 1386, 2982, 5874, 10791, 18733, 31031, 49413, 76076, 113764, 165852, 236436, 330429, 453663, 612997, 816431, 1073226, 1394030, 1791010, 2277990, 2870595, 3586401, 4445091, 5468617, 6681368, 8110344, 9785336

Offset

1,2

Comments

Partial sums of A024166. Third partial sums of cubes (A000578).

Antidiagonal sums of the array A213564. - Clark Kimberling, Jun 18 2012

Links

Danny Rorabaugh, Table of n, a(n) for n = 1..10000

C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Dead link]

C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Cached copy, May 15 2013]

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Formula

This sequence could be obtained from the general formula n*(n+1)*(n+2)*(n+3)*...*(n+k)*(n*(n+k)+(k-1)*k/6)/((k+3)!/6) at k=3. - Alexander R. Povolotsky, May 17 2008

G.f. -x*(1+4*x+x^2) / (x-1)^7. - R. J. Mathar, Dec 06 2011

Sum_{n>0} 1/a(n) = (8/3)*(25-9*sqrt(5)*Pi*tan(sqrt(5)*Pi/2)). - Enrique Pérez Herrero, Dec 02 2014

a(k) = MagicNKZ(3,k,7) where MagicNKZ(n,k,z) = Sum_{j=0..k+1} (-1)^j*binomial(n+1-z,j)*(k-j+1)^n. (Cf. A101104.) - Danny Rorabaugh, Apr 23 2015

Mathematica

s1=s2=s3=0; lst={}; Do[s1+=n^3; s2+=s1; s3+=s2; AppendTo[lst, s3], {n, 0, 6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 15 2009 *)
Table[n*(n + 1)*(n + 2)*(n + 3)*(1 + 3*n + n^2)/120, {n, 31}] (* Michael De Vlieger, Apr 20 2015 *)

Prog

(PARI) a(n)=sum(l=1, n, sum(j=1, l, sum(m=1, j, sum(i=m*(m+1)/2-m+1, m*(m+1)/2, (2*i-1))))) \\ Alexander R. Povolotsky, May 17 2008
(Sage) [n*(n+1)*(n+2)*(n+3)*(1+3*n+n^2)/120 for n in range(1, 32)] # Danny Rorabaugh, Apr 20 2015
(Magma) [n*(n+1)*(n+2)*(n+3)*(1+3*n+n^2)/120 : n in [1..35]]; // Vincenzo Librandi, Apr 23 2015

Crossrefs

Cf. A000537, A101097, A101102, A101104.

Sequence in context: A323039 A211614 A244497 * A187693 A200529 A289255

Adjacent sequences: A101091 A101092 A101093 * A101095 A101096 A101097

Keyword

nonn,easy

Author

Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004

Extensions

Edited by Ralf Stephan, Dec 16 2004

Status

approved